The total number of E-O-E linkages (a bridging oxygen atom bonded to two atoms of the same element E, where E = P or S, and not counting any O-O…

Draw the connectivity of each oxo-compound. $\mathrm{H_4P_2O_7}$ (pyrophosphoric acid) has two $\mathrm{PO_4}$ tetrahedra sharing one corner oxygen, so it has 1 $\mathrm{P{-}O{-}P}$ link. $\mathrm{H_2S_2O_7}$ (disulfuric acid, oleum) has two $\mathrm{SO_4}$ units sharing one oxygen, so it has 1 $\mathrm{S{-}O{-}S}$ link. $\mathrm{H_2S_2O_8}$ (peroxodisulfuric acid) joins the two sulfur atoms through a peroxide bridge $\mathrm{S{-}O{-}O{-}S}$, which is excluded, so it contributes 0. $\mathrm{H_4P_2O_6}$ (hypophosphoric acid) has a direct $\mathrm{P{-}P}$ bond and no bridging oxygen, so it contributes 0. $\mathrm{P_4O_{10}}$ has the adamantane-like cage with six $\mathrm{P{-}O{-}P}$ edges and four terminal $\mathrm{P{=}O}$ groups, so it contributes 6. Total $=1+1+0+0+6=8$.